Biomedical Engineering Reference
In-Depth Information
the Donnan equilibrium. The discovery of this chloride shift phenomenon is attrib-
uted to Hamburger HJ, which is popularly called Hamburger's chloride shift. An
ion exchange transporter protein in the cell membrane facilitates the chloride shift.
A build up of H
+
ions in the red blood cell would also prevent further conversion
and production of the bicarbonate ion. However, H
+
ions bind easily to reduced
hemoglobin, which is made available when oxygen is released. Hence, free H
+
ions
are removed from the solution. Reduced hemoglobin is less acidic than oxygenated
hemoglobin. As a result of the shift of Cl
−
ions into the red cell and the buffering
of H
+
ions onto reduced hemoglobin, the intercellular osmolarity increases slightly
and water enters causing the cell to swell. The reverse process occurs when the red
blood cells pass through the lung.
3.2.3 Goldman Equation
Most biological components contain many types of ions including negatively
charged proteins. However, the Nernst equation is derived for one ion after all per-
meable ions are in Donnan equilibrium. The membrane potential experimentally
measured is often different than the Nernst potential for any given cell, due to the
existence of other ions. An improved model is called the
Goldman equation
(also
called the Goldman-Hodgkin-Katz equation), named after American scientist Da-
vid E. Goldman, which quantitatively describes the relationship between membrane
potential and permeable ions. According to the Goldman equation, the membrane
potential is a compromise between various equilibrium potentials, each dependent
on the membrane permeability and absolute ion concentration. Assuming a planar
and infinite membrane of thickness (
L
) with a constant membrane potential (
ΔΦ
m
),
d
dx
Φ
ΔΦ
(3.8)
=
m
L
Substituting (3.8) into (3.4) and rearranging gives
−
dC
dx
=
(3.9)
J
z
ΔΦ
total
+
m
C
D
L
AB
Equation (3.9) is integrated across the membrane with the assumption that
total flux is constant and
C
L
. Furthermore, con-
sidering the solubility of the component using (3.14), and rearranging an equation
to the total flux is obtained as:
=
C
i
when
x
=
0,
C
=
C
o
when
x
=
⎡
⎛
zF
RT
ΔΦ
⎞
⎤
m
−
⎜
⎟
⎝
⎠
−
P F
ΔΦ
⎢
C Ce
−
⎥
mem
m
i
o
J
=
ln
(3.10)
⎢
⎥
total
RT
zF
RT
ΔΦ
⎛
⎞
m
−
⎢
⎥
⎜
⎟
⎝
⎠
1
−
e
⎣
⎦
